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Question

When a stationary wave is formed, then its frequency is


A
Same as that of the individual waves
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B
Twice that of the individual waves
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C
Half that of the individual waves
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D
That of the individual waves
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Solution

The correct option is A Same as that of the individual waves
Let us consider a progressive wave of amplitude $$a$$ , wavelength $$\lambda$$ and frequency $$f=\dfrac { 1 }{ T } $$ travelling in the direction of $$X$$- axis whose equation is given by-
$${ y }_{ 1 }=a\sin { (2\pi [\dfrac { t }{ T } -\dfrac { x }{ \lambda  } ] } )$$
This wave is reflected from a free end and it travels in the negative direction of $$X$$- axis with frequency $$f=\dfrac { 1 }{ T } $$
$${ y }_{ 1 }=a\sin { (2\pi [\dfrac { t }{ T } +\dfrac { x }{ \lambda  } ] } )$$
According to principle of superposition
$$\Rightarrow { y }_{ 1 }+{ y }_{ 2 }=a\sin { (2\pi [\dfrac { t }{ T } -\dfrac { x }{ \lambda  } ] } )+a\sin { (2\pi [\dfrac { t }{ T } +\dfrac { x }{ \lambda  } ] } )$$
$$\Rightarrow { y }_{ 1 }+{ y }_{ 2 }=a\sin { \left( \dfrac { 2\pi t }{ T }  \right)  } \cos { \left( \dfrac { 2\pi x }{ \lambda  }  \right)  } $$
Here we can see the frequency of resultant wave is also $$f=\dfrac { 1 }{ T } $$.

Physics

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